Finding matrices of perturbation using creation/annihilation operators

In summary: The matrix ##\hat{W}## is a potential energy, and it doesn't have a ground state at all because the potential does not have a lower limit (it can be as negative as you want if you increase ##x## or ##y## enough).
  • #1
Keru
20
1
"Given a 3D Harmonic Oscillator under the effects of a field W, determine the matrix for W in the base given by the first excited level"

So first of all we have to arrange W in terms of the creation and annihilation operator. So far so good, with the result:

W = 2az2 - ax2 - ay2 + 2az+ 2 -ax+ 2 -ay+ 2 + 2{az, az+}) - {ax, ax+}) - {ay, ay+}

As for the following part, I've proceeded like this, which I'm pretty sure it's wrong at some point.

In terms of the ladder operator, I'm using the basis:
|nx, ny,nz>

With ni corresponding to the level of excitement on a certain coordinate. I have taken the first excited level to be:

|1,0,0>

I generally find the matrix elements like:

Mi,j = <i|M^|j>

So my thought has been, I have to find them in the basis |1,0,0>, let's just:

<1,0,0|W|1,0,0>

Which looks terrible because that's clearly a 1x1 "matrix".
What am I missing? Do the matrix actually correspond to |1,0,0>, |0,1,0>, |0,0,1> since all these possible combinations correspond to the first energy state? Is that simply not the basis I should be using?

Any help would be greatly appreciated.
 
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  • #2
Where is this from and what's the operator ##\hat{W}(\hat{x},\hat{y},\hat{z})## in terms of the position operators ##\hat{x},\hat{y},\hat{z}## ?

A single energy eigenstate (like the first excited level) obviously can't be a basis for the system, you need an infinite number of states.
 
  • #3
hilbert2 said:
Where is this from and what's the operator ##\hat{W}(\hat{x},\hat{y},\hat{z})## in terms of the position operators ##\hat{x},\hat{y},\hat{z}## ?

A single energy eigenstate (like the first excited level) obviously can't be a basis for the system, you need an infinite number of states.

In terms of position operators it is:
W = 2z2 - x2 - y2
and it comes from the electric cuadrupole.

So, there's something I'm not getting right then. I'll try to translate the exercise as precisely as possible:
"Determine the matrix of W in the base given by the states of the first energy excitation". Should I do all in terms of position operators and forget about creation/annihilation operators? The previous section asked us to express it in terms of them, but this one does not explicitly tell us to use them, so maybe it's just nonsensical trying to do it like i was?
Sorry if the question seems obvious I am pretty much new at this.
 
  • #4
If the ##\hat{W} = 2\hat{z}^2 - \hat{x}^2 - \hat{y}^2## is supposed to be a potential energy (you first have to multiply it with something of dimensions [energy]/[length]^2 to make it have dimensions of energy), then it doesn't have a ground state at all because the potential does not have a lower limit (it can be as negative as you want if you increase ##x## or ##y## enough). Let's see what others say about this.
 
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Related to Finding matrices of perturbation using creation/annihilation operators

1. What are creation and annihilation operators?

Creation and annihilation operators are mathematical operators used in quantum mechanics to describe the creation and destruction of particles. They are represented by symbols a† and a, respectively, and are used to manipulate quantum states.

2. How are creation and annihilation operators related to matrices of perturbation?

Creation and annihilation operators are used to construct matrices of perturbation in quantum mechanics. These matrices represent the changes in the quantum state caused by perturbations, such as external forces or interactions with other particles.

3. What is the significance of finding matrices of perturbation?

Finding matrices of perturbation is crucial in understanding the behavior of quantum systems under external influences. These matrices allow scientists to calculate the probabilities of different outcomes and make predictions about the behavior of quantum particles.

4. What are some applications of matrices of perturbation in scientific research?

Matrices of perturbation are used in a variety of research areas, such as quantum computing, quantum chemistry, and particle physics. They are also used in the development of new technologies, such as quantum sensors and quantum communication devices.

5. How are matrices of perturbation calculated using creation and annihilation operators?

The process of calculating matrices of perturbation using creation and annihilation operators involves representing the perturbation as a sum of operators, then using the commutation and anti-commutation relations to simplify the expression. The resulting expression can then be used to construct the matrix of perturbation.

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